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Affine Invariant Interacting Langevin Dynamics for Bayesian Inference

Garbuno-Inigo, Alfredo and Nüsken, Nikolas and Reich, Sebastian (2020) Affine Invariant Interacting Langevin Dynamics for Bayesian Inference. SIAM Journal on Applied Dynamical Systems, 19 (3). pp. 1633-1658. ISSN 1536-0040. doi:10.1137/19m1304891.

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We propose a computational method (with acronym ALDI) for sampling from a given target distribution based on first-order (overdamped) Langevin dynamics which satisfies the property of affine invariance. The central idea of ALDI is to run an ensemble of particles with their empirical covariance serving as a preconditioner for their underlying Langevin dynamics. ALDI does not require taking the inverse or square root of the empirical covariance matrix, which enables application to high-dimensional sampling problems. The theoretical properties of ALDI are studied in terms of nondegeneracy and ergodicity. Furthermore, we study its connections to diffusion on Riemannian manifolds and Wasserstein gradient flows. Bayesian inference serves as a main application area for ALDI. In case of a forward problem with additive Gaussian measurement errors, ALDI allows for a gradient-free approximation in the spirit of the ensemble Kalman filter. A computational comparison between gradient-free and gradient-based ALDI is provided for a PDE constrained Bayesian inverse problem.

Item Type:Article
Related URLs:
URLURL TypeDescription Paper
Garbuno-Inigo, Alfredo0000-0003-3279-619X
Reich, Sebastian0000-0002-5336-8904
Additional Information:© 2020 SIAM. Published by SIAM under the terms of the Creative Commons 4.0 license. Received by the editors December 6, 2019; accepted for publication (in revised form) by M. Wechselberger April 29, 2020; published electronically July 16, 2020. This research was partially supported by Deutsche Forschungsgemeinschaft (DFG, German Science Foundation) through grants SFB 1294/1 318763901 and SFB 1114/2 235221301. The work of the first author was supported by the generosity of Eric and Wendy Schmidt by recommendation of the Schmidt Futures program, by Earthrise Alliance, by the Paul G. Allen Family Foundation, and by the National Science Foundation (grant AGS-1835860). We would like to thank Christian Bär, Andrew Duncan, Franca Hoffmann, Andrew Stuart, and Jonathan Weare for valuable discussions related to the sampling methods proposed in this paper.
Funding AgencyGrant Number
Deutsche Forschungsgemeinschaft (DFG)SFB 1294/1 318763901
Deutsche Forschungsgemeinschaft (DFG)SFB 1114/2 235221301
Schmidt Futures ProgramUNSPECIFIED
Earthrise AllianceUNSPECIFIED
Paul G. Allen Family FoundationUNSPECIFIED
Subject Keywords:Langevin dynamics, interacting particle systems, Bayesian inference, gradient flow, multiplicative noise, affine invariance, gradient-free
Issue or Number:3
Classification Code:AMS subject classifications: 65N21, 62F15, 65N75, 65C30, 90C56
Record Number:CaltechAUTHORS:20201029-154635142
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:106348
Deposited By: Tony Diaz
Deposited On:29 Oct 2020 23:02
Last Modified:16 Nov 2021 18:53

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